Real Analysis

Determine if the series $\frac{\cos(n\pi)}{n^3}$ is absolutely convergent, conditionally convergent, or divergent.

Determine if the series $\frac{(-1)^{n+1}}{\sqrt[3]{n}}$ is absolutely convergent, conditionally convergent, or divergent.

Determine if the series $\frac{(-1)^{n}n^3}{n^3 + 5}$ is absolutely convergent, conditionally convergent, or divergent.

Given a series $\sum a_n$, determine whether it converges absolutely or conditionally. Specifically, consider $\sum (-1)^{n-1}/\sqrt{n}$ and evaluate its convergence using the alternating series test and discuss its absolute convergence.

Find all of the accumulation points of the sequence $a_n$ where $a_n = (1 + (-1)^n, 2^n + (-2)^n)$.

Prove that the sequence definition and the neighborhood definition of limit points are equivalent.

Consider the series $\sum_{k=1}^{\infty} (-1)^{k+1} \frac{1}{k}$. Prove that this alternating harmonic series converges using the Alternating Series Test.

Consider the series $\sum_{k=1}^{\infty} (-1)^{k+1} \frac{k+3}{k(k+1)}$. Determine if this series converges using the Alternating Series Test.

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$ converges or diverges using the alternating series test.

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n-1}$ converges or diverges using the alternating series test.

How do you calculate functions like $e^x$, $ext{sin}(x)$, or $ext{cos}(x)$ at any given $x$ value?

What are the uses of non-analytic smooth functions in real analysis?

Prove the Weierstrass approximation theorem, which states that for any continuous function defined on a closed interval, there exists a sequence of polynomials that converges uniformly to the function on that interval.

Given a continuous function on the interval [0, 1], show that Bernstein polynomials converge uniformly to the function as n goes to infinity.

Define an injective map and a surjective map.

Show that a map from the natural numbers to the square numbers, defined by mapping $x$ to $x^2$, is bijective.

Prove the Bolzano-Weierstrass theorem: Every bounded sequence has a convergent subsequence.

Consider whether the converse of the Bolzano-Weierstrass theorem is true: If a sequence has a convergent subsequence, then it is bounded. Give an example of an unbounded sequence that has a convergent subsequence.

Prove that any bounded sequence has a subsequence that converges.

Given a sequence, determine if it is both bounded and monotonic. If it is, prove that it converges.